Your friend is right that GDP and similar measures are flawed—or, at least, incomplete. But this isn't exactly news to economists!
For example, Richard Easterlin was arguing in the 1970s that GDP is a poor measure of national well-being. The specific issues with GDP that you point out in your question (failure to account for the value of innovations or for free products, etc.) are so well-known that they now make it into tutorials aimed at high school students. For a more recent take on the issue, here is a discussion paper on the topic of technology and GDP measurement (the author also has a book on the topic of GDP as a measure).
To be a little bit more concrete about how economists know the measure is flawed, let's look at how economic theory helps us to set objectives and guides us in measuring our performance in a simple example.
Economics as a body of knowledge suggests variables of interest to a policymaker. Suppose, for the sake of illustration, we are concerned only with maximising peoples' material well-being. We know that we can get a measure of material welfare by computing the area bounded by the general equilibrium demand curve and supply function. So, in an ideal world, we would just measure this total surplus and pursue policies that increase it. For a variety of reasons, this quantity is difficult to measure—especially at the aggregate level over time.
The 'next best' alternative is to use the economic model to tell us what variables influence this surplus, and then try to measure those variables instead. Obvious examples of such variables are income (which expands demand possibilities) and technology (which expands production possibilities). Broadly speaking, if income is going up and technology is improving then the surplus (which is what we are actually interested in) should also be increasing.
It is important to keep an eye on both variables because the theory tells us that surplus can rise as income falls if, for example, technology is improving fast enough (just think about the extreme case where technology is so good that we can produce everything costlessly; then price and income would be zero, but surplus from consumption would nevertheless be very high).
With this framework in mind, it is very natural to look at GDP and say: "This is a measure of income, which is a part of the picture. But it is missing another component (technology) that we know, from theory, is also important."
If measures like GDP are known to be so flawed, why are they used? It is important to understand that no serious economist cares about GDP per se. Instead, economists stress the importance of a portfolio of policies that work together to improve overall well-being of the populace. There are many factors that weigh on well-being. These include income and productivity (which will show up in GDP), but also technological innovation, environmental improvement, improved health outcomes, reduced inequality, and so on.
Given these objectives, some way is needed to measure the efficacy of policy. Because the agenda is multi-faceted, we need enough indicators to measure the various dimensions we are interested in. GDP is useful because it is a widely-available measure of one part of what economists are interested in (material wellbeing). But this should not be confused with a situation where GDP is the objective in its own right. Nor should it be thought that economists have no interest in improving upon this measure.
The monomaniacal fixation with GDP is really a trait of the media and politics rather than practicing economists.